[Proof of Proposition ]
We follow the proof in [1]}.
Consider dataset \(S={_1,_2,\ldots , _n}\) , \(_i \in {-\frac{1}{\sqrt{d}},\frac{1}{\sqrt{d}}}^d\)  - the dataset is therefore constrained to lie in the unit Euclidean ball. Consider the following function \(f(,) = -{}{}\)  with the constraint set \(\)  being the unit Euclidean ball. It is easy to see that \(f(\cdot ,)\)  is 1-Lipschitz for all \(\) . The empirical risk becomes \(\hat{F}_S() = -{}{\frac{1}{n}\sum _{i=1}^n_i}\) , the minimum of which over the unit ball is \(_S^* = \frac{\frac{1}{n}\sum _{i=1}^n_i}{{\frac{1}{n}\sum _{i=1}^n_i}}\) .
